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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

-3
votes
0answers
42 views

Function satisfying $f(x+y)=f(x)f(y),~\forall x,y \in \mathbb R$ [on hold]

Let $f:\mathbb R \rightarrow \mathbb R$ be a function satisfying $f(x+y)=f(x)f(y),~\forall x,y \in \mathbb R$ and $\lim_{x \rightarrow 0}f(x)=1$,then prove that $f(x)=a^x$ for some non zero $a \in \...
0
votes
1answer
66 views

Fundamental Thm of Calculus proof (with the chain rule) - please help me

It might be a easy question for you :) Set the two variables as $u $ and $v$ $s.t.$ $u=g(x)$ and $v=h(x)$ Let the differentiable function $f(x,t)$ and its antiderivative $F(x)(=F(u,v,x))=\int_{u}^{...
0
votes
1answer
23 views

Is this set “not closed”? [duplicate]

Is it correct to say that this set $E=(0,1]$ where $E\subseteq R$ (Where $R$ is the set of real numbers) is not closed?
0
votes
1answer
34 views

Probability of singletons in with an uncountable sample space

Let us assume that we have a sample space $\Omega=[0,1]$. Why is it not possible to have all the singletons $\{x\}\in\Omega$ with non-zero probability measure. I searched for an answer on this forum ...
-5
votes
1answer
40 views

If $f:U\subseteq\mathbb{R}^2\to\mathbb{R}$, $\partial f/\partial x$, $\partial f/\partial y$ exist and are continuous, then $f$ is differentiable [on hold]

Prove: If $f: U\subseteq \mathbb{R}^2\to\mathbb{R}$ and $\partial f/\partial x$, $\partial f/\partial y$ exist and are continuous on $U$, then $f$ is differentiable on $U$
2
votes
1answer
32 views

Prove harmonic function inequality with the Mean-Value Property

Let $\Omega \subset \mathbb{R}^n$ open and let $u$ be a harmonic function in $\Omega.$ If $K \subset \Omega$ is compact, then prove $$ \sup\limits_{x \in K} |u(x)| \le \frac{n}{\omega_n ~dist(K,...
2
votes
1answer
62 views

Uniform convergence of $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^2}{(1 + x^2)^n}$ on $\Bbb{R}$.

I was trying to prove that $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^2}{(1 + x^2)^n}$ converges uniformly on $\Bbb{R}$. was trying to use $M$ test, in which I am trying to bound $|f_{n}(x)| < M_{n} $ ...
5
votes
1answer
57 views

Why is the interchange of a sum and an integral not justified here?

This problem arose when I was studying the analytic continuation of the $\Gamma$ function. Consider $$f(z) = \int_{0}^1 e^{-t}t^{z-1} dt$$ This integral converges only for $\Re(z)>0$. However, ...
2
votes
1answer
47 views

If every open cover of a set $F\in\mathbb{R}$ admits a finite subcover, then $F$ is closed and bounded.

This should be a pretty known proof in analysis for many of you but I've (I think) proved it in a different way than I've seen in any textbook. The bounded proof is pretty straightforward and I ...
2
votes
1answer
28 views

Positive measurable set contains some number pair

Let $E \in [0,1]$ be a Lebesgue measurable set with positive measure. Show that there exists $a,b$ such that all three numbers $a$, $a+b$, $a+2b$ are contained in $E$. I know that if no such $a,b$, ...
1
vote
3answers
67 views

Why can I not put the limit inside the limit definition of $e$?

We know $e:= \lim_{n\to \infty}(1+\frac{1}{n})^n$ We also know $x^n$ is continuous. Why is there a contradiction in the following? $e=\lim_{n\to \infty}(1+\frac{1}{n})^n=(\lim_{n\to \infty}(1+\frac{...
1
vote
1answer
35 views
+100

Propagation of regularity for the heat equation

Let $\Omega$ be an open bounded subset of $\mathbb R^N$. Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega)$ and consider the following boundary value problem for the heat equation:...
4
votes
1answer
88 views

Let $f(x)$ be continuous from $[0, +\infty)$ to $ [0, +\infty)$, and $\int_{0}^{+\infty}f(x)dx$ diverges.

Prove there exists some $a>0$ such that the series $\sum_nf(an)$ diverges. I think it can be useful to partition $[0, +\infty)$ on $[n, n+1)$ and choose some $a_n$ for every $n$. But I can't ...
1
vote
1answer
22 views

Biharmonic equation

I know the gradient $\Delta f = \sum_{i=1}^n \frac {\partial f}{\partial x_i}$ Then looking at the Laplace operator $\Delta^2 f = \sum_{i=1}^n \frac {\partial^2f}{\partial x^2_i}$ Now my first ...
0
votes
1answer
29 views

Rationals can be the set of continuity of a function? [duplicate]

Most of the functions that I have seen have their discontinuities on rationals and continuities on irrationals! I am wondering if there is any exampe of some function whose continuities are rationals?...
3
votes
2answers
82 views

Proof that a sequence converges.

Let $(a_n)_{n=1}^{\infty}$ be a sequence, let $\; 0 < q < 1$ such that $\;\forall n \in \mathbb{N}$ $\;n>2$ $ \left | a_{n+1} -a_n \right | < q\left | a_{n} -a_{n-1}\right |$ prove ...
0
votes
1answer
31 views

Closed formula for the general term of a sequence (using factorials)

Given the sequences $a_{n+2}=\frac{2a_{n}}{n+2}$ I am trying to find a general way of expressing this in terms of $a_{k}$ I keep getting $a_{k}=\frac{a_{0}}{\left(\frac{k}{2}\right)!}$ but I am told ...
2
votes
1answer
46 views

Limit of $ \frac{f(x) f''(x)}{(f'(x))^{2}}$ at $\alpha$ with $f(\alpha)= f'(\alpha)= f''(\alpha)=0$ and $f'''(\alpha) \not= 0$ via L'hopital

Suppose $f$ is a $C^{4}$ function with $f(\alpha)= f'(\alpha)= f''(\alpha)=0$ and $f'''(\alpha) \not= 0$. I wish to compute the following limit: $$ \lim_{x \to \alpha} \frac{f(x) f''(x)}{(f'(x))^{2}...
2
votes
1answer
44 views

Subset notation in Rudin's “Principles of Mathematical Analysis”

I've just began reading Walter Rudin's "Principles of Mathematical Analysis". Early on in the text is introduced subset notation, with which I am familiar. My problem is that Rudin doesn't seem to ...
2
votes
1answer
51 views

Let : $P(x)=x^{3}+ax^{2}+bx+c$ where $(a,b,c)\in Z^3$

Question : If : $P(x)=x^3+ax^2+bx+c$ where $(a,b,c)\in Z^3$ And $m,n,k$ root of $P(x)$ such that : $m.n=k$ Then show that : $2P(-1)$ multiple of $P(1)+P(-1)-2[1+P(0)]$ My try : We known ...
1
vote
1answer
34 views

Convert a real range to an integer range.

Let's say that we have a set of integers in the range $[1, 4]$. Now, I have a function that will calculate a distance between two vectors, and this function returns a real number in the range $[0, 1]$....
1
vote
2answers
32 views

Convergence of indicator functions series

Say $E$ is a measurable set, and $\{E_k\}$ is a series of measurable sets defined by $$ E_k \subset E, m(E \setminus E_k) < \frac{1}{k}, k = 1, 2, 3, ... $$ Do their corresponding indicator ...
1
vote
1answer
29 views

A question about Abel Discriminance in series

I just read the proof of Abel discriminance in the series chapter. It says If ∑bn converges, and {an} is monotone with |an|≤K, then series ∑anbn converges. in my ...
0
votes
0answers
50 views

Calculate the sum.

There is given positive decreasing sequence of real numbers $a_{1},a_{2},a_{3},...$ Calculate the sum in nice form: $$SUM=\sum_{i=1}^{N}a_{i}-2\sum_{1\le i<j\le N}a_{i}a_{j}$$ If it is impossible ...
1
vote
0answers
24 views

A ball contained in the image of a diffeomorphism

I am trying to prove the following. If $\varphi: U \subset \mathbb{R}^n \to V \subset \mathbb{R}^n$ is a diffeomorphism such that for $x\in B(0,r) \subset U$, $$ |\varphi(x)-x| < \epsilon|x|, $$ ...
1
vote
1answer
17 views

Some help with a measure-theoretic integration problem.

Let $g: [0, \infty) \rightarrow \mathbb{R}$ be continuous, increasing, and bounded with $g(0)= 0$ and $g(x) > 0$ for $x > 0$. Let $(X, \mathcal{A}, \mu)$ be a finite measure space and $f_{n}: X \...
2
votes
1answer
30 views

$X$ is compact Hausdorff. Then ring automorphism $\mathcal C(X)\to \mathcal C(X)$ preserves norm.

Consider $X$ a compact Hausdorff space. Suppose $\sigma:\mathcal C(X)\to\mathcal C(X)$ is a ring automorphism and $\Bbb C$-algebra or $\Bbb R$-algebra where $\mathcal C(X)$ denotes all continuous map ...
0
votes
1answer
37 views

Finding the supremum and infimum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$

$\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$ What is the supremum and infimum of the above set? The set is $\left\{1, 2^{\frac{1}{2}}, 3^{\frac{1}{3}},....\right\}$ Now, $n^{\frac{...
0
votes
3answers
39 views

Show bijectivity of $f:(-1,1)\rightarrow \mathbb{R}, f(x)=\frac{x}{1-|x|}$

Show bijectivity of $f:(-1,1)\rightarrow \mathbb{R}, f(x)=\frac{x}{1-|x|}$ So in order to show injectivity $f(a)=f(b) \Rightarrow a=b$ so $\frac{a}{1-|a|}=\frac{b}{1-|b|}$. But how do I prove that? ...
-1
votes
2answers
22 views

Check if a set is open or close or neither

I've got the set G = {$x\in R^2: |x| \le 2, |y-1| \gt 1$} I need to determine whether this set is open or closed, I've drawn the set and I think it is neither, but I'm not sure how to proof it. ...
0
votes
2answers
35 views

limit involving harmonic function

Let $u$ harmonic function in $\mathbb{R}^3 -\{0\}$. I know that $$\lim_{x\to0} \sqrt{|x|} \cdot u(x)=k< \infty$$ I'm trying to show that $k=0$. I tried by contradiction, but I failed and I'm ...
0
votes
0answers
31 views

Is $p\neq 0$, then there are $x_+, x_- \in \mathbb{C}$ with $x_\pm ^2=p$.

Is $p\neq 0$, then there are $x_+, x_- \in \mathbb{C}$ with $x_\pm ^2=p$. My proof: The proof for $p>0$ is trivial, since $\mathbb{R}\subset \mathbb{C}$ and it's true $\forall p\in \mathbb{R}$. ...
5
votes
2answers
98 views

Evaluate the indefinite integral of multiplication of two functions

Let $f,g:[a,+\infty]\to\mathbb{R}$ be continuous and $K>0$ s.t. $$ \left|\int_c^df(x)\,dx\right|\leq K \quad \forall c,d\in [a,\infty). $$ If $g\in C^1$ is decreasing with $\displaystyle\lim_{x\...
3
votes
0answers
38 views

Calculus of variation: quasi convex function

I am given the following things: let $M\in \mathbb{R}^{(N\times n)\times(N\times n)}$ be a symmetric matrix. The function $f: \mathbb{R}^{N\times n} \to \mathbb{R}$ is defined as: $$f(\xi):=\langle M\...
4
votes
1answer
87 views

If $f: [1,\infty)\to [e,+\infty)$ is increasing, $\int_1^\infty \frac{dx}{f(x)}=+\infty$, show that $\int_1^\infty \frac{dx}{x\ln f(x)}=+\infty$. [on hold]

If $f: [1,\infty)\to [e,+\infty)$ is increasing and $$\int_1^\infty \frac{dx}{f(x)}=+\infty$$show that $$\int_1^\infty \frac{dx}{x\ln f(x)}=+\infty$$ How can I show this?
1
vote
0answers
47 views

Prove that the function is increasing

Let $g : \mathbb{N} \rightarrow \mathbb{R}_+$ with $g(0) = 0$. Let's start by assuming that there exists positive constants $\Gamma_-\leq 1 \leq \Gamma_+$ such that $$\Gamma_- \leq g(k)-g(k-1)\leq \...
-1
votes
0answers
55 views

Inequality for $a+b+c=3$ $3\geq (ab)^{bc}+(bc)^{ca}+(ca)^{ab}$

It's a new problem that I find interesting : Let $a,b,c>0$ such that $a+b+c=3$ then we have : $$3\geq (ab)^{bc}+(bc)^{ca}+(ca)^{ab}$$ My try : If $ab\leq 1$ , $bc\leq 1$ , $ca\leq 1$ the ...
0
votes
0answers
30 views

Wrong usage of Gauss Theorem?

In a book, where the author wants to proof that $- \frac{1}{4\pi}\Delta \frac{1}{|r-r'|}$ has the filtering property of the $\delta$-distribution $$ \int_V - \frac{1}{4\pi}\Delta \frac{1}{|r-r'|} \...
1
vote
1answer
25 views

Cantor's Intersection Theorem iff complete metric spaces?

If a metric space is complete, we know that the Cantor's Intersection Theorem holds. Does the converse also hold? And if not, what is a suitable counterexample for the same? Also, if the converse ...
4
votes
1answer
48 views

Proof continuous function $g$ defined on $[0,1]$ has a fixedpoint $x \in [0,1]$ [duplicate]

Claim: For the continuous function $g:[0,1]\rightarrow[0,1] \exists x \in [0,1]: g(x_1) = x_1$ Proof: Case 1: If g(0) = 0 and/or g(1) = 1 then g has at least one fix point on one of the ends of the ...
0
votes
2answers
49 views

Let $f: [0,10) \to [0,10] $ be a continuous function then which is correct

Let $f: [0,10) \to [0,10] $ be a continous map then (a) $f$ need not have any fixed point (b) $f$ has atleast $10$ fixed point (c) $f$ has atleast $9$ fixed point (d) $f$ has atleast one fixed ...
0
votes
1answer
68 views

Consider a function $f: \Bbb R \to \Bbb R$ s.t $|f(x)-f(y)|\leq 4321|x-y|$. Choose the correct option:

This is a MCQ question, consider a function $f: \Bbb R \to \Bbb R$ s.t $|f(x)-f(y)|\leq 4321|x-y|$. Choose the correct option: 1) $f$ is always diffrentiable. 2) there exist atleast one such $f$ ...
1
vote
4answers
98 views

$x_1=\sqrt{2}$ and we have $x_{n+1}=(\sqrt{2})^{x_n}$. Find out the limit. [duplicate]

Suppose we have $x_1=\sqrt{2}$ and we have $x_{n+1}=(\sqrt{2})^{x_n}$. We have to find out the limit of the sequence. I was observing that $x_2=(\sqrt{2})^\sqrt{2}$ and $x_3=(\sqrt{2})^{{\sqrt{2}}^{...
0
votes
1answer
41 views

Real analytic function is uniquely determined by its derivatives at $0$

In Klenke's Probability Theory one can read [...] $f$ can be expanded about any point $t\in \mathbb R$ in a power series with radius $\geq R$. In particular it is analytic and is hence determined ...
0
votes
1answer
23 views

Show that $ \int_0^1 \sup_{n\in\mathbb N}\{ f_n(x) \}dm(x)=\infty $

Let $m$ denote Lebesgue measure on $[0,1]$. Let $\{f_n\}$ be a sequence of Lebesgue measurable functions on $[0,1]$ with values in $[0,\infty]$ such that $$ \lim_{n\to\infty} f_n(x)=0 $$ almost ...
0
votes
1answer
26 views

Is it possible to apply the Dirichlet's uniform convergence test

For a series of functions given by $$\sum_{k=1}^{\infty}\frac{1}{k}\sin\left(\frac{x}{k+1}\right)$$ on some bounded nonempty set $A$ in $\mathbb{R}$, is it possible to apply the Dirichlet's uniform ...
0
votes
1answer
20 views

negative semidefinite matrix

I got a positive definite matrix $B$, that is, $V(x)=x^TBx>0$ for any vector $x≠0$. I am clear with the statement that $λ_\min∥x∥_2^2≤V(x)≤λ_\max∥x∥_2^2$ for any $x≠0$, where $λ_\min$ and $λ_\max$ ...
0
votes
2answers
58 views

Way to explain what the limit is [on hold]

Can anyone come up with a good way to explain why or way to show that for $a$ and $b$ any real numbers we have that $$ -(x-a)^2-b \log(x) + x \rightarrow - \infty $$ as $x$ tends to infinity along ...
2
votes
3answers
102 views

A non-constant function satisfies $f(x)f(\frac{1}{x})=1$

Let $f:A\rightarrow \mathbb R,~A \subseteq \mathbb R$ be a non-constant function satisfies $f(x)f\big(\frac{1}{x}\big)=1$, then can we say that $f(x)=\pm x^n$ for some $n\in \mathbb N$?
-3
votes
0answers
24 views

Convergent and point-wise convergent of a sequence [on hold]

Show that the sequence $(f_n)_n$ of functions given by $f_n(x)=\tan^{−1}(nx)$ for $x\in [0,\infty)$ is uniformly convergent on any interval $[a,b]$, $a,b>0$ but is only pointwise convergent on $[0,...